On the Fast Computation of the Weight Enumerator Polynomial and the $t$ Value of Digital Nets over Finite Abelian Groups

• Published in: SIAM journal on discrete mathematics Vol. 27; no. 3; pp. 1335 - 1359
• Main Authors: Dick, Josef, Matsumoto, Makoto
• Format: Journal Article
• Language: English
• Published: Philadelphia Society for Industrial and Applied Mathematics 01.01.2013
• Subjects: Algorithms; Applied mathematics; Arithmetic; Computation; Construction costs; Digital; Linear algebra; Mathematical analysis; Mathematical models; Polynomials; Rings (mathematics); Weight reduction
• ISSN: 0895-4801; 1095-7146
• DOI: 10.1137/120893677

In this paper we introduce digital nets over finite abelian groups which contain digital nets over finite fields and certain rings as a special case. We prove a MacWilliams-type identity for such digital nets. This identity can be used to compute the strict $t$-value of a digital net over finite abelian groups. If the digital net has $N$ points in the $s$-dimensional unit cube $[0,1)^s$, then the $t$-value can be computed in $\mathcal{O}(N s \log N)$ operations and the weight enumerator polynomial can be computed in $\mathcal{O}(N s (\log N)^2)$ operations, where operations mean arithmetic of integers. By precomputing some values the number of operations of computing the weight enumerator polynomial can be reduced further. [PUBLICATION ABSTRACT]